U.S. Pat. No. 4,960,422 discloses a hair removing device having a detachable epilator head that can be separated from a grip housing for cleaning the head, or for replacement with an analogous shaving head. A rotary motor is incorporated in the housing to provide a driving force of epilating or shaving the hairs. Due to the detachable nature of the device, the housing also includes a gear which is driven by the motor and is responsible for detachable meshing engagement with an associated gear provided in the epilator head or the shaving head. That is, the epilator head is drivingly connected to the motor by meshing engagement of the gears when the epilator head is attached to the housing. With this gear connection, however, the device suffers from a strict dimensional tolerance between the gears, i.e., a distance between the axes of the gears for successfully transmitting the driving force from the motor to movable members in the head, which in turn requires a correspondingly strict detachable connection between the housing and the detachable head. Further, when the head is operating under a heavy load, the gear connection is likely to transmit vibrations occurring in the heavy load operation of the head directly to the housing, thereby generating harsh vibrations which detract from a comfortable operation by the user grasping the housing.